Let G be a connected semisimple group over ℚ. Given a maximal compact subgroup K⊂G(ℝ) – such that X=G(ℝ)/K is a Riemannian symmetric space – and a convenient arithmetic subgroup Γ⊂G(ℚ), one constructs an arithmetic manifold S=S(Γ)=Γ∖X. If H⊂G is a connected semisimple subgroup such that H(ℝ)∩K is maximal compact, then Y=H(ℝ)/H(ℝ)∩K is a symmetric subspace of X. For each g∈G(ℚ) one can construct an arithmetic manifold S(H,g)=(H(ℚ)∩g-1Γg)∖Y and a natural immersion jg:S(H,g)→S induced by the map H(𝔸)→G(𝔸),h↦gh. Let us assume that G is anisotropic, which implies that S and S(H,g) are compact. Then, for each positive integer k, the map jg induces a restriction map
Rg:Hk(S,ℂ)→Hk(S(H,g),ℂ). In this paper we focus on symmetric spaces associated to the unitary and orthogonal groups, namely O(p,q) and U(p,q), and give explicit criterions for the injectivity of the product of the maps Rg (for g running through G(ℚ)) when restricted to the strongly primitive (in the sense of Vogan and Zuckerman) part of the cohomology. We also give explicit criterions for the injectivity of the map
Hk(S(H),ℂ)→Hk+ dim S- dim S(H)(S,ℂ) dual to the restriction map
dual to the restriction map Re.
The results we obtain fit into a larger conjectural picture that we describe and which bare a strong analogy with the classical Lefschetz Theorems. This may sound quite surprising that such an analogy still exists in the case of the real arithmetic manifolds.
We reduce the global problems mentioned above to local ones by using Theorems of Burger and Sarnak and isolations properties of cohomological representations in the automorphic dual. The methods used then are mainly representation-theoretic.
We finally derive some applications concerning the non vanishing of some cohomology classes in arithmetic manifolds.